The equation of the ellipse whose vertices are $( \pm 5,\;0)$ and foci are $( \pm 4,\;0)$ is

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If $ e_1, e_2 $ be their eccentricities respectively, then  $e_1^{ - 2} + e_2^{ - 2}$ equals

The equation $\frac{{{x^2}}}{{2 - r}} + \frac{{{y^2}}}{{r - 5}} + 1 = 0$ represents an ellipse, if

If a tangent having slope of $ - \frac{4}{3}$ to the ellipse $\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{{32}} = 1$ intersects the major and minor axes in points $A$ and $B$ respectively, then the area of $\Delta OAB$ is equal to .................. $\mathrm{sq. \, units}$ ($O$ is centre of the ellipse)

Find the equation of the ellipse whose vertices are $(±13,\,0)$ and foci are $(±5,\,0)$.

Find the equation for the ellipse that satisfies the given conditions: Vertices $(0,\,\pm 13),$ foci $(0,\,±5)$.